A Radical Characterization of Abelian Varieties

2017/08/25 by Theodore Hui, Hui, Theodore
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1708.07899

openalex publication_date 2017/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a square-free abelian variety defined over a number field K. Let S be a density one set of prime ideals \mathfrakp of O_K. A famous theorem of Faltings says that the Frobenius polynomials P_A,\mathfrakp(x) for \mathfrakp∈ S determine A up to isogeny. We show that the prime factors of |A(\mathbbF_\mathfrakp)|=P_A,\mathfrakp(1) for \mathfrakp∈ S also determine A up to isogeny over an explicit finite extension of K. The proof relies on understanding the ℓ-adic monodromy groups which come from the ℓ-adic Galois representations of A, and the absolute Weyl group action on their weights.

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